Evaluate all infinite series that appear in your formula

Assignment Help Mathematics

Reference no: EM132481235

Question 1: Consider ARMA(1,1) model X_t = ΦX_t-1 + Z_t + θZ_t-1, where |Φ| < 1, θ ∈ R and the random variables Z_t are i.i.d. with mean zero and variance σ²_Z.

(a) Use the linear representation (see Example 2.9 in Lecture Notes) to obtain the general formula for yx(h), h ≥ 0. Evaluate all infinite series that appear in your formula.

Question 2: Consider AR(1) model X_t = ΦX_t-1+ Z_t, where |Φ|< 1 and the random variables Z_t are i.i.d. with mean zero and variance σ²Z.

(a) Apply the Durbin-Levinson algorithm to get Φ₁₁, Φ₂₁, Φ₂₂.
(b) Try to prove that Φ_nn = 0 for all n ≥ 2 (recall that Φ_nn is partial autocovariance at lag n).

Question 3. In the following question you will simulate data from AR(1) model and do basic analysis. Please provide relevant codes and relevant numerical answers, but do not include pictures. I give several hints, while the usage of commands ar and arima can be consulted on the course webpage.

(a) Simulate n = 200 observations from AR(1) with Φ = 0.8 and σ²_z = 1. You can plot for yourself the ACF and PACF to see that the generated data really come from AR(1), but please do not include plots.

Hint: Use arima.sim(list (order = c (1,0,0) , ar = 0.8) , n = 200)

(b) Use R to estimate the parameters Φ and σ²_z using the Yule-Walker method.
Hint: Use ar

Hint: Section 5.1 in Lecture Notes.

(d) Use R to estimate the parameters Φ and σ²_z using the MLE method.

Hint: Use ar.

(e) Use estimates either from (b) or (d) to predict X₂₀₁.

Hint: You need to read the last observation from your data set. Remember about the mean!

(f) In (d), why R chose AR(1), not AR(2)?

Hint: Use arima and compare AIC for both models.

Question 4. In the following question you will use the data set priceNSW.txt available on the course webpage. Please provide relevant codes and relevant numerical answers, but do not include pictures.

(a) Fit an ARMA model.
Hint: You decide on the method

(b) Use R to estimate all model parameters (for example, if your model chosen is AR(2), estimate Φ₁, Φ₂ and σ²_z.

Hint: You need to read the last observation from your data set. Remember about the mean!

Attachment:- Assignment Dataset.rar

Reference no: EM132481235

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